The Impact of Higher-Order Interactions on the Synchronization of Hindmarsh–Rose Neuron Maps under Different Coupling Functions

Abstract

In network analysis, links depict the connections between each pair of network nodes. However, such pairwise connections fail to consider the interactions among more agents, which may be indirectly connected. Such non-pairwise or higher-order connections can be signified by involving simplicial complexes. The higher-order connections become even more noteworthy when it comes to neuronal network synchronization, an emerging phenomenon responsible for the many biological processes in real-world phenomena. However, involving higher-order interactions may considerably increase the computational costs. To confound this issue, map-based models are more suitable since they are faster, simpler, more flexible, and computationally more optimal. Therefore, this paper addresses the impact of pairwise and non-pairwise neuronal interactions on the synchronization state of 10 coupled memristive Hindmarsh–Rose neuron maps. To this aim, electrical, inner linking, and chemical synaptic functions are considered as two- and three-body interactions in three homogeneous and two heterogeneous cases. The results show that through chemical pairwise and non-pairwise synapses, the neurons achieve synchrony with the weakest coupling strengths.

Synchronization in simplicial complexes of memristive Rulkov neurons

“…In an adjacency matrix, each element represents the presence or absence of a connection between a pair of nodes in a network, while in an adjacency tensor, each element represents the presence or absence of a connection between a set of nodes of arbitrary size (Lucas et al, 2020 ). For example, indicates that the i -th and j 1 -th nodes are connected through a link while shows that nodes with indices i, j 1 , ..., j d combine to form a polyhedron ( d -simplex; Mirzaei et al, 2022 ; Parastesh et al, 2022 ; Mehrabbeik et al, 2023 ). Furthermore, G (1) , ..., G ( d ) are the coupling functions describing the type of interactions among 2, ..., d +1 units building simplicial complexes.…”

“…To demonstrate the impact of adding three-, four-, and five-node chemical interactions stepwise on the synchronization state, they took into account a fully connected network of five conventional Rulkov maps with two-node diffusive inner linking and chemical connections. Mehrabbeik et al ( 2023 ) recently investigated the synchronization of different higher-order networks made up of 10 Hindmarsh-Rose maps. The authors intended to figure out the impact of higher-order synaptic functions applied to two-node and three-node communication on network global synchronization.…”